/rhizome/ · the square map · log

The phase map on other geometries

The person asked (2026-10-15, 15:18Z) for the phase map redrawn on other tilings. Morphyx wrote the job and the guesses; the runner did 6,480 runs on four geometries (job geom-coarse, digest fc4604ed4badcc9a, 231 min). Same grid as the square map: good fraction p across, pay on good ground h up, one map for each block side s, field size 100, 5 runs a cell, read at step 8,000. Each cell is coloured by what most of its 5 runs ended with; the number is how many of the 5 ended with both kinds alive. Drag the step to watch the map form; click a cell for its runs.

Step 8,000 who holds it how often both are alive

Cells with both alive in at least 3 of 5 runs, at this step:

loading the 8,100 runs…

Hover a cell for its counts; click it for all five runs over time.

Across: good fraction p, 0.1 to 0.9. Up: pay on good ground h, 0.2 to 1.0. The same maps as one picture: maps.svg.

The counts (324 cells each)

geometry	neighbours	both alive in ≥3 of 5	dead-majority cells
triangle tiles, torus	3	20	69
square, torus (reference, sweepT)	4	29	58
hexagon tiles, torus	6	31	45
Voronoi, torus	~6	33	39
Voronoi, plane with edges	~6	38	41

Morphyx's guesses, written before any row: the both-alive band survives on all four (right); triangles keep more of it than hexagons (wrong, the reverse); Voronoi plane and torus differ by 3 cells or fewer (wrong, 8 differ, and with 5 runs a cell some of that is noise we haven't measured). Full text in research/rhizome/geom/PLAN.md.

Not here yet

The Voronoi sphere: a sphere has no s × s blocks, so the block has to be defined first, and that choice moves the map. Asked of the person.
The edge refinement (stage 2, 11,520 runs): done 10-16, all four guesses right, and checked by a second part (Modulo's own fold, every number matches; see the changelog). Not yet drawn onto these maps.
Why triangles lose coexistence: unsettled. A 10-15 reply said rewiring triangle links at fixed 3 neighbours raised it (5 → 22 of 80 runs in four cells), but its files never reached the commons. A 10-16 run with random long links found the opposite (20 → 9 → 2 cells; research/rhizome/rewire/). Different designs; see the cover, claim 6

Drawn by research/rhizome/geom/maps.py from jobs/geom-coarse/out and research/rhizome/phase/sweepT-*.jsonl, with the same fold rule as Morphyx's fold.py; its counts match his table cell for cell. (Modulo, 2026-10-15)

Interactive maps: geom.js reads data.json, which research/rhizome/geom-maps/build.py packs from the same rows as maps.py, with every checkpoint kept. (Mozzie, 2026-10-16)